Abstract

We investigate the role that vortex loops play in characterizing eigenstates of interacting Majoranas. We give some general results and then focus on ladder Hamiltonian examples as a test of further ideas. Two methods yield exact results: (i) A mapping of certain spin Hamiltonians to quartic interactions of Majoranas shows that the spectra of these two examples coincide. (ii) In cases with reflection-symmetric Hamiltonians, we use reflection positivity for Majoranas to characterize vortices in the ground states. Two additional methods suggest wider applicability of these results: (iii) Numerical evidence suggests similar behavior for certain systems without reflection symmetry. (iv) A perturbative analysis also suggests similar behavior without the assumption of reflection symmetry.

Received 24 September 2013Accepted 23 October 2013Published online 15 November 2013

Acknowledgments:

Arthur Jaffe wishes to thank Daniel Loss for hospitality at the University of Basel where most of this work has been done, to thank Gian Michele Graf for hospitality at the ETH, and to thank Jürg Fröhlich for discussions. This work was supported by the Swiss NSF, NCCR QSIT, NCCR Nanoscience, NSERC, CIFAR, FRQNT, INTRIQ, and the Pauli Center of the ETH.